Abstract
This paper is devoted to the estimation of a partial graphical model with a structural Bayesian penalization. Precisely, we are interested in the linear regression setting where the estimation is made through the direct links between potentially high-dimensional predictors and multiple responses, since it is known that Gaussian graphical models enable to exhibit direct links only, whereas coefficients in linear regressions contain both direct and indirect relations (due e.g. to strong correlations among the variables). A smooth penalty reflecting a generalized Gaussian Bayesian prior on the covariates is added, either enforcing patterns (like row structures) in the direct links or regulating the joint influence of predictors. We give a theoretical guarantee for our method, taking the form of an upper bound on the estimation error arising with high probability, provided that the model is suitably regularized. Empirical studies on synthetic data and a real dataset are conducted.
Highlights
We are interested in the recovery and estimation of direct links between high-dimensional predictors and a set of responses
We have two goals in this paper: 1. Give some theoretical guarantees to the model introduced in Chiquet et al [7]. 2
Our work is a generalization of [29], using the same technical tools to establish an upper bound on the estimation error when a prior on the direct links generates an additional structural penalty in the objective, provided that the model is suitably regularized
Summary
We are interested in the recovery and estimation of direct links between high-dimensional predictors and a set of responses. Whereas the graphical models seem a natural way to go, we propose to take account of a prior knowledge on the predictors, when possible This is typically the case when dealing with genetic markers whose joint influence may be anticipated thanks to some kind of genetic distance, or when the predictors are supposed to represent a continuous phenomenon so that consecutive covariates probably act together. In this regard, while taking up the graphical approach, we introduce some Bayesian information in a structural regularization of the estimation procedure, the inference remains frequentist, thereby following the idea of Chiquet et al [7]. Before introducing the model and the organization of this work, let us describe the notation used throughout the paper
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